Uniqueness of solutions for an elliptic equation modeling MEMS. (English) Zbl 1171.35044

Esposito, Pierpaolo; Ghoussoub, Nassif

Methods Appl. Anal. 15, No. 3, 341-354 (2008).

Let \(\Omega\) be an open set in \({\mathbb R}^N\) and let \(f\) be a smooth nonnegative function defined on \(\Omega\). This paper is devoted to the study of solutions \(0<u<1\) of the nonlinear elliptic equation \(-\Delta u=\lambda f(x)/(1-u)^2\) in \(\Omega\), under the Dirichlet boundary condition \(u=0\) on \(\partial\Omega\). The authors are mainly concerned with the role of the real parameter \(\lambda\), in relationship with the geometry of the domain \(\Omega\). There are established several existence and non-existence results and the proofs combine various tools, including the implicit function theorem and the Lyapunov-Schmidt reduction theorem.

Reviewer: Teodora-Liliana Rădulescu (Craiova)

MSC:

35J60	Nonlinear elliptic equations
35B32	Bifurcations in context of PDEs
35D10	Regularity of generalized solutions of PDE (MSC2000)
35J20	Variational methods for second-order elliptic equations
35Q60	PDEs in connection with optics and electromagnetic theory

Keywords:

MEMS; stable solutions; quenching branch